Estimating additional resistivity by permeability in brine-saturated sandstones

Wenjun Zhao; Tangyan Liu; Jizhou Tang; Jie Zhang

doi:10.1144/petgeo2022-067

Abstract

Determining rock resistivity for saturation estimation in reservoirs is challenging due to the complex nature of pores in the rock. This paper aims to establish a computational relationship between formation factors (F) and permeability (K) by combining theoretical and experimental data. Firstly, the relationship between the permeability of the curved capillary model and formation factors, as well as the relationship between the permeability of the complex curved capillary model and formation factors, are deduced. Theoretical analysis proved that the formation factors (F) have a power relationship with permeability (K) and porosity (⁠ $φ$ ⁠), and confirms the existence of additional resistivity (R_x). To validate the theoretical study, we conducted a model analysis using open experimental data from 35 sandstone cores with different porosities and permeabilities from the tight gas sandstone in the Western US basins, which measured resistivity data in saline at 20, 40 and 80 Kppm, respectively. We confirmed the existences of additional resistivity (R_x) by fitting the relationship between the rock resistivity of saturated formation water (R₀) and the formation water resistivity (R_w). We then fitted the formation resistivity change factor (F_d) with permeability (K), the formation resistivity change factor (F_d) with porosity (⁠ $φ$ ⁠), the additional resistivity (R_x) with permeability (K), and the additional resistivity (R_x) with porosity (⁠ $φ$ ⁠). Both the changeable formation resistivity change factor (F_d) and additional resistivity (R_x) showed a strong linear relationship with permeability (K) in logarithmic coordinates.

We also verified the existence of a suitable equation using available experimental data by changing the formation parameters and permeability. The study shows that the fitting equations may be utilized to determine the changeable formation resistivity change factor (F_d), additional resistivity (R_x) and the rock resistivity of saturated formation water (R₀) with varying permeability. The predicted rock resistivity of saturated formation water (R₀) strongly correlates with that measured in the laboratory, providing better precision for future reservoir evaluations using saturation estimations.

In the field of petroleum exploration, accurately determining the porosity, permeability and water saturation of a reservoir is critical for conducting effective evaluation. Archie's formula (1942), developed by G.E. Archie, is a widely used method for evaluating saturation through resistivity analysis of experimental data. These parameters are influenced primarily by reservoir geology, including depositional conditions and genesis. An evaluation of water content saturation is one of the most critical elements in reservoir evaluation. Archie proposed a model for quantitatively evaluating saturation by resistivity through comprehensive analysis of experimental data (Archie 1942). Many other equations have been derived using Archie's formulation as the basic equation. Archie (1947) used an equation to measure the fluid saturation (S_w) of the rock using porosity (⁠ $Φ$ ⁠), the cementation coefficient (m), the saturation index (n), resistivity (R_w), the true formation resistivity of the rock (R_t) and the tortuosity factor (a). After Archie's formula was introduced, more academics such as Waxmanand Smits (1968), Waxman and Thomas (1974), Roberts and Schwartz (1985), Kumar et al. (2011) and Nazemi et al. (2019) conducted in-depth investigations using experimental and theoretical analyses to understand the rock conduction mechanism. They discovered that several factors, such as pore geometry, shale volume and roughness of the pore wall, influence the formation factor (F). The relationships of F v. $Φ$ and resistivity index (RI) v. S_w are not always clearly linear in a double logarithmic coordinate system. The parameters in Archie's formula are related to the formation lithology and pore structure, and they vary with the salinity of the formation, and the surrounding pressure and temperature. However, in low-porosity and low-permeability reservoirs in particular, the traditional Archie formula seems ineffective. Waxman and Smits (1968) discovered the additional conductivity of clay minerals and proposed their Waxman–Smits (W–S) model based on cation-exchange capacity calculation. Clavier et al. (1977) divided the conductive layer on the surface of clay minerals into the surface layer close to the clay minerals that is affected by the composition of the clay minerals and the free electron layer away from the clay minerals that is unaffected by clay, and developed an improved W–S model. As the understanding of reservoir genesis has deepened, the study of the saturation model based on the conductive mechanism of complex pore structures has gradually become a hotspot in reservoir evaluation. Shang et al. (2003) proposed an equivalent rock component saturation model that divided the pores and throat-forming pore space into parallel and series pores, defined the pore structure coefficient and saturation scale factor, and provided a clear parameter calculation method that effectively improved the saturation calculation accuracy for reservoirs with a complex pore structure. Xiao et al. (2013) classified the physical characteristics of different capillary pressure curve morphological types according to their porosity range, and established a statistical relationship between the ‘J function’ and saturation (Xiao et al. 2013; Nazemi et al. 2021; Tavakoli et al. 2022). Numerous scholars, both domestic and international, have conducted significant research on resistivity models and established various models suitable for sandstone, including the effective medium resistivity model, which is entirely composed of conductive minerals, the pure conduction model of rock, and the skeleton composed of part of the conductive particles of the conductive pure rock conductive model (Hunt 2004; Montaron 2009; Liu et al. 2013; Revil et al. 2013; Tang et al. 2015). Different algorithms have been proposed for different saturation models (Glover et al. 2010; Wang and Liu 2017). However, all the existing resistivity models face a challenge when describing the conductivity pattern of muddy sandstone with partial conductive minerals in the skeleton.

Zhang et al. (2020) derived equations for the resistivity change factor (F_d) of rocks and additional resistivity (R_x) without making any assumptions. The validity of these equations was tested using thousands of experimental data points and previously published data (Zhang et al. 2020).

Archie proposed a functional expression between the formation factor and porosity but noted the need to clarify the relationship between permeability and the formation factor compared to porosity. However, he did not conduct further analysis on this issue. Permeability is one of the most important parameters required in reservoir characterization. When there is no physical or chemical change in the pore fluid and porosity of the rock, the value for permeability and the formation factor is determined by the pore structure.

Geological setting

This study utilizes open experimental data from tight gas sandstone in the western USA to examine the relationship between core and log petrophysical properties and lithofacies sedimentary characteristics in Mesaverde Group tight gas sandstones. The study examines 40 cores in the Washakie, Uinta, Piceance, Greater Green River, Wind River and Powder River basins (see Fig. 1). The shaly intervals of the Mesaverde Group are dominated by mudstones and burrowed silty shales, as well as lenticular and wavy-bedded very shaly sandstones, and wavy-bedded to ripple cross-laminated shaly sandstones. The sandstone intervals are dominated by ripple cross-laminated, cross-bedded, planar-laminated and massive, very-fine- to fine-grained sandstones (Byrnes et al. 2008).

Fig. 1.

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The locations of wells sampled in the study (Byrnes et al. 2008).

This paper utilized a collection of open experimental data from tight gas sandstone in the western USA for study. To gain insight into the internal pore structure connectivity of the cores in the two basins, we first analysed the core images (Figs 2–5).

Fig. 2.

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The core images for Green River C-47TIP Top Shallow (depth, 2729.8 ft; porosity, 19.4%; permeability, 8.02 mD; environment of deposition, marine shoreface). Characteristics: the rock contains intergrain pores with good connectivity. Porosity consists entirely of sparsely distributed, poorly connected microporosity within the interparticle voids of the mudstone and shale matrix. Cementation is present and includes minerals such as siderite, ferroan calcite and pyrite. Organic matter can also be found in some areas of the rock.

Fig. 3.

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The core images for Green River C-47TIP Top Shallow (depth, 2754.7 ft; porosity, 20.7%; permeability, 1.29 mD). Characteristics: weak pore connectivity with intergranular dissolution pores. The porosity is primarily composed of poorly connected primary and secondary intergranular mesopores, sparse moldic pores and quartz overgrowth cement. The amount of quartz cement is sparse.

Fig. 4.

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The core images for Piceance MV 33-34 (depth, 6579.8 ft; porosity, 5.3%; permeability, 0.000247 mD). Characteristics: the rock exhibits intergranular pores with poor pore connectivity. Its porosity mainly comprises sparsely distributed and poorly connected intergranular micropores filled with clay. The amount of quartz cement is prominent, and ferroan calcite is sparse.

Fig. 5.

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The core images for Piceance M30-2-96 (depth, 6379.5 ft; porosity, 3.8%; permeability, 0.000303 mD). Characteristics: the rock exhibits intergranular pores with poor pore connectivity. The porosity consists entirely of sparse, poorly connected microporosity located within interparticle voids of the mudstone and shale matrix. The cement present in the rock includes siderite, ferroan calcite and pyrite.

Analysing core images from the same area can also aid in comprehending the impact of pore structure on interconnectivity. It is evident from the core images that the connectivity between the two basins differs significantly. The Green River Basin exhibits well-developed intergranular dissolution pores with good pore connectivity and permeability, while the Piceance Basin is mainly characterized by isolated intergranular pores with poor pore connectivity and low permeability.

Methodology

Archie's extensive experimental data revealed that the formation factor, F, and porosity, $Φ$ ⁠, could be approximated effectively and modelled as a straight line in logarithmic coordinates. The exponent m value, according to Archie, typically ranges from 1.3 to 2. The coefficients in Archie's formula are derived from a specific physical model and adjusted through real rock sample experiments. However, if a region's rock samples have complex lithological characteristics, the experimentally derived parameters may not satisfy engineering needs. Hence, researchers need to modify the values of m and n when employing Archie's formula in various regions with distinct lithologies. Therefore, the coefficients in Archie's formula may differ depending on the region.

Shaly sandstones can present difficulties in logging analysis due to the presence of clays with a high cation-exchange capacity (CEC) and formation brines with a low salinity (Kukal et al. 1983). To address these challenges, several algorithms have been proposed to calculate the water saturation in shaly sands. These include the empirical Simandoux model (1963), and the more theoretical dual water and Waxman–Smits models (Waxman and Smits 1968; Waxman and Thomas 1974). These models attempt to account for the influence of clay minerals and low-salinity brines on the electrical resistivity of the formation, and the subsequent measurements obtained from logging tools.

In earlier studies, various conductive models were used by Waxman and Smits (1968), Clavier et al. (1984) and Silva and Bassiouni (1988) to investigated the additional conductance of clay-exchange cations and to assess saturation with fixed and variable parameters of Archie's equation.

Zhang et al. (2020) discussed the additional resistivity and conductivity of natural reservoirs. They demonstrated that any core (R₀) resistivity could be expressed as a function of the equivalent additional resistivity (R_x), F_d and R_w, as shown in :

R_{0} = F_{d} R_{w} + R_{x}

(1)

where R_x is the additional resistivity (⁠

Ω

·m) and F_d is formation resistivity change factor (dimensionless) (see Table 1).

Table 1.

Nomenclature

R₀

Rock resistivity of fully saturated brine (⁠

Ω

·m)

R_w

Rock fluid resistivity (⁠

Ω

·m)

R_t

Rock resistivity (⁠

Ω

·m)

R_x

Additional resistivity (⁠

Ω

·m)

m and n

Empirical constants

F_d

Formation resistivity change factor

K

Permeability (mD)

Δ

P

Pressure difference between the two ends of the capillary tube (10⁵ Pa)

Q

Flow rate (cm³ s⁻¹)

μ

Viscosity of water (10⁻³ Pa s).

C_w

Water conductivity (mhos)

a

Archie cementation intercept

Relationship between the permeability of the curved capillary model and the formation factor

In this paper, we further explore the relationship between the formation factors of porous media and porosity or permeability using a theoretical approach based on a simple model. We begin by developing a simple model, as shown in Figure 6, which assumes that the core is a single pore structure filled with movable fluid. Based on the bending capillary model, the pores in the rock are simplified to a single bending capillary with radius r. The tortuosity of the curved capillary is defined as

τ_{w}

⁠, and

L_{w}

is the actual length of the capillary channel, while

L_{0}

is the straight length of the capillary channel. L_w is expressed in as:

L_{w} = τ_{w} L_{0} .

(2)

Fig. 6.

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Schematic diagram of the bent capillary model.

The water-bearing volume is then:

V_{w} = π r_{w}^{2} L_{w} .

(3)

According to Poiseuille's law, the flow rate of water in a curved hairy bundle is obtained as follows:

Q = \frac{r_{w}^{2}}{8} \frac{Δ P π r_{w}^{2}}{μ L_{w}} \times 10^{- 8}

(4)

where Q is the flow rate (cm³ s⁻¹),

Δ

P is the pressure difference between the two ends of the capillary tube (10⁵ Pa) and

μ

indicates the viscosity of water (10⁻³ Pa s).

According to Darcy's linear seepage law, the seepage flow, Q, can be expressed as:

Q = \frac{K Δ P π R^{2}}{μ L_{0}} .

(5)

After rewriting and (), and () can be obtained:

\frac{r_{w}^{2}}{8} \frac{Δ P π r_{w}^{2}}{μ L_{w}} \times 10^{- 8} = \frac{K Δ P π R^{2}}{μ L_{0}}

(6)

K = \frac{r_{w}^{4}}{8 τ_{w} R^{2}} \times 10^{- 8} .

(7)

In this model, only the formation water in rock pores is considered to conduct electricity in order to construct an equivalent pure rock model. According to Ohm's law, the following can be obtained:

R_{o} \frac{L_{0}}{π R^{2}} = R_{w} \frac{L_{t}}{π r_{w}^{2}}

(8)

where R₀ is the rock resistivity of fully saturated brine only (⁠

Ω

·m), R_w is fluid resistivity (⁠

Ω

·m) and R_t is rock resistivity of fully saturated brine and gas (⁠

Ω

·m).

The formation factor, F, can then be expressed as:

F = \frac{R_{o}}{R_{w}} = \frac{L_{t}^{2}}{L_{0}^{2}} \frac{π R^{2} L_{0}}{π r_{w}^{2} L_{t}} = \frac{τ^{2}}{φ} .

(9)

After rewriting and (), can be obtained:

F = \frac{1}{64} \frac{r_{w}^{8}}{R^{4}} φ^{- 1} K^{- 1} \times 10^{16} .

(10)

can then be simplified to:

F = c (φ K)^{- 1}

(11)

where c represents the dimensionless coefficient related to the pore structure. shows the exponential relationship between the formation factor, permeability and porosity in a single capillary model.

Relationship between permeability and the formation factor in a complex curved capillary model

A conductive pathway can be formed in mud with a conductive skeleton due to the presence of conductive minerals in the rock, dispersed clay and residual movable water in micropores. Thus, two conductive pathways in pores would exist in parallel. This model is illustrated in Figure 7.

Fig. 7.

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Model diagram of the complex curved capillary and its electrical conductivity diagram.

Similarly, based on the curved capillary model, the length of the curved capillary can be defined as L_x and the tortuous degree as

τ_{x}

⁠, and subsequently

L_{x} = τ_{x} L_{0}

⁠, so the volume in this pathway is:

V_{x} = π r_{x}^{2} L_{x} .

(12)

The volume of residual water in this pathway is then:

V_{x w} = π r_{x}^{2} L_{x} - n \frac{3}{4} π r_{x}^{3} .

(13)

The core porosity is

φ

and the bound water saturation is

S_{w i}

⁠, giving:

V_{x w} = π R^{2} L_{0} φ S_{w i} .

(14)

By combining with , we can obtain:

\begin{aligned} π R^{2} L_{0} φ S_{w i} & = π r_{x}^{2} L_{x} - n \frac{3}{4} π r_{x}^{3} \\ (π L_{x} - n \frac{3}{4} π r_{x}) r_{x}^{2} & = π R^{2} L_{0} φ S_{w i} . \end{aligned}

(15)

According to , this equation can be solved iteratively to obtain the value of

r_{x}

⁠. According to Poiseuille's law, the flow rate of water in the curved hairy bundle is obtained as follows:

Q = \frac{Δ P π r_{w}^{4}}{8 μ L_{w}} \times 10^{- 8} + \frac{Δ P π r_{x}^{4}}{8 μ L_{x}} \times 10^{- 8}

(16)

where Q is the flow rate (cm³ s⁻¹),

Δ

P is the pressure difference between the two ends of the capillary tube (10⁵ Pa) and

μ

indicates the viscosity of water (10⁻³ Pa s).

According to Darcy's linear seepage law, the seepage flow, Q, can be expressed as:

Q = \frac{K Δ P π R^{2}}{μ L_{0}} .

(17)

By combining simultaneous and , can be obtained as follows:

\begin{aligned} \frac{Δ P π r_{w}^{4}}{8 μ L_{w}} \times 10^{- 8} + \frac{Δ P π r_{x}^{4}}{8 μ L_{x}} \times 10^{- 8} = \frac{K Δ P π R^{2}}{μ L_{0}} \\ K = (\frac{r_{w}^{4}}{8 τ_{w} R^{2}} + \frac{r_{x}^{4}}{8 τ_{x} R^{2}}) \times 10^{- 8} . \end{aligned}

(18)

According to the parallel conduction theory:

\frac{1}{R_{0} (L_{0} / π R^{2})} = \frac{1}{R_{w} (L_{w} / π r_{w}^{2})} + \frac{1}{R_{x} (L_{x} / π r_{x}^{2})}

(19)

where R_w is the rock fluid resistivity (⁠

Ω

·m) and R_x is additional resistivity (⁠

Ω

·m).

can be simplified to:

R_{x} \frac{L_{x}}{r_{x}^{2}} R_{w} \frac{L_{w}}{r_{w}^{2}} = R_{0} \frac{L_{0}}{R^{2}} (R_{x} \frac{L_{x}}{r_{x}^{2}} + R_{w} \frac{L_{w}}{r_{w}^{2}}) .

(20)

By defining the effective porosity,

φ_{e}

⁠, as follows:

φ_{e} = \frac{π r_{w}^{2} L_{w}}{π R^{2} L_{0}}

(21)

then:

R_{0} = \frac{R_{x} R_{w} (L_{x} / r_{x}^{2}) (τ^{2} / φ_{e})}{R_{x} (L_{x} / r_{x}^{2}) + R_{w} (L_{w} / r_{w}^{2})} .

(22)

The formula can be simplified to:

\begin{aligned} R_{0} & = R_{w} \frac{τ^{2}}{φ_{e}} \frac{R_{x} (L_{x} / r_{x}^{2})}{R_{x} (L_{x} / r_{x}^{2}) + R_{w} (L_{w} / r_{w}^{2})} \\ = R_{w} \frac{τ^{2}}{φ_{e}} \frac{1}{1 + ((R_{w} L_{w} r_{x}^{2}) / (R_{x} L_{x} r_{w}^{2}))} \\ R_{0} & = R_{w} \frac{τ^{2}}{φ_{e}} (1 - \frac{((R_{w} L_{w} r_{x}^{2}) / (R_{x} L_{x} r_{w}^{2}))}{1 + ((R_{w} L_{w} r_{x}^{2}) / (R_{x} L_{x} r_{w}^{2}))}) \\ R_{0} & = \frac{τ^{2}}{φ_{e}} R_{w} + D R_{x} \end{aligned}

(23)

then:

R_{0} = F_{d} R_{w} + D R_{x}

(24)

where F_d is the formation resistivity change factor (dimensionless) and D denotes the coefficient related to the pore structure (dimensionless).

Equation (24) shows that the rock resistivity (R₀) can be expressed as a primary function of the formation resistivity change factor F_d and the additional resistivity R_x when fully saturated with brine in the complex curved capillary model . The formation resistivity change factor F_d is influenced by the effective porosity and tortuosity of the rock.

By combining simultaneous , () and () and solving them, F_d can be obtained as follows:

F_{d} = \frac{τ^{2}}{φ_{e}} = a K^{- b}

(25)

where a and b represent the dimensionless coefficient related to the pore structure. A power relationship exists between permeability and the formation factor and between porosity and the formation factor in complex curved capillary models.

Results

We selected 35 core samples from two basins with significantly different pore connectivity (Tables 2 and 3).

Table 2.

Experimental data of rock resistivity in the Green River Basin

				Brine salinity (Kppm):		20			40			80
				C_w (mhos):		3.02			5.73			10.5
				R_w (⁠ $Ω$ ·m):		0.331			0.17452			0.09524
USGS Library No.	Basin	Well name	Depth (ft)	Routine porosity (%)	In situ Klinkenberg gas permeability (mD)	R₀/R_w	1/R₀	m, a = 1	R₀/R_w	1/R₀	m, a = 1	R₀/R_w	1/R₀	m, a = 1
T204-2	Green River	B-2A Spider Creek	9063	15.2	206	36.9	0.0818	1.77	37.4	0.1531	1.77	36.4	0.2884	1.76
R780-1		C-47 Tip Top Shallow	2699.8	21.2	28.2	19.7	0.1536	1.83	19.8	0.2896	1.84	20.6	0.5099	1.86
E712-1		B-54 Big Piney	3431.9	17.5	15.7	27.9	0.1083	1.8	30.6	0.187	1.85	32.4	0.3244	1.88
T204-1		B-2A Spider Creek	9022.9	12.1	10.2	56.3	0.0537	1.82	61.8	0.0928	1.86	62.4	0.1682	1.87
R780-4		C-47 Tip Top Shallow	2845.5	22	6.34	16.3	0.1859	1.76	17.1	0.3348	1.8	18.2	0.5771	1.83
R780-2		C-47 Tip Top Shallow	2754.7	20.7	1.29	22.2	0.1362	1.85	22.3	0.2565	1.85	22.8	0.4597	1.87
T204-3		B-2A Spider Creek	9098	6.6	0.0188	94.5	0.032	1.58	136.8	0.0419	1.68	163.8	0.0641	1.74
T195-1		5 Pinedale	12 159.6	9	0.0111	137.6	0.022	1.93	167	0.0343	2	193.4	0.0543	2.06
E894-2		1 Old Road	11 927.8	11.3	0.00925	70	0.0432	1.89	74	0.0775	1.91	93.7	0.1121	2.02
E712-2		B-55 Big Piney	3480.8	8.9	0.0058	76.8	0.0393	1.75	106.6	0.0538	1.88	119.9	0.0876	1.92
B029-3		A-1 Wasp	11 443.8	6.6	0.00178	298.3	0.0101	2.05	329.2	0.0174	2.08	382.6	0.0274	2.14
SHELL Vible		Vible 1D-11D	12 518.5	5.9	0.00172	215.1	0.014	1.85	270.3	0.0212	1.93	339.8	0.0309	2
E894-1		1 Old Road	11 897.3	5.2	0.000995	152.6	0.0198	1.65	222.1	0.0258	1.78	292.4	0.0359	1.87
B029-1		A-1 Wasp	11 332.9	3.6	0.000832	258.4	0.0117	1.63	327.8	0.0175	1.7	443.5	0.0237	1.79
R780-3		C-47 Tip Top Shallow	2759.9	9.1	0.000708	87.3	0.0346	1.84	101.7	0.0563	1.91	142.1	0.0739	2.04
T195-2		5 Pinedale	12 161.5	6.5	0.000361	249.4	0.0121	1.99	298.2	0.0192	2.06	361.4	0.0291	2.13
B029-4		A-1 Wasp	11 534	1.5	0.000137	195.2	0.0155	1.21	280.6	0.0204	1.29	351	0.0299	1.35
B029-2		A-1 Wasp	11 374.9	0.5	0.000018	301.8	0.01	0.99	490.1	0.0117	1.08	753.8	0.0139	1.15

				Brine salinity (Kppm):		20			40			80
				C_w (mhos):		3.02			5.73			10.5
				R_w (⁠ $Ω$ ·m):		0.331			0.17452			0.09524
USGS Library No.	Basin	Well name	Depth (ft)	Routine porosity (%)	In situ Klinkenberg gas permeability (mD)	R₀/R_w	1/R₀	m, a = 1	R₀/R_w	1/R₀	m, a = 1	R₀/R_w	1/R₀	m, a = 1
T204-2	Green River	B-2A Spider Creek	9063	15.2	206	36.9	0.0818	1.77	37.4	0.1531	1.77	36.4	0.2884	1.76
R780-1		C-47 Tip Top Shallow	2699.8	21.2	28.2	19.7	0.1536	1.83	19.8	0.2896	1.84	20.6	0.5099	1.86
E712-1		B-54 Big Piney	3431.9	17.5	15.7	27.9	0.1083	1.8	30.6	0.187	1.85	32.4	0.3244	1.88
T204-1		B-2A Spider Creek	9022.9	12.1	10.2	56.3	0.0537	1.82	61.8	0.0928	1.86	62.4	0.1682	1.87
R780-4		C-47 Tip Top Shallow	2845.5	22	6.34	16.3	0.1859	1.76	17.1	0.3348	1.8	18.2	0.5771	1.83
R780-2		C-47 Tip Top Shallow	2754.7	20.7	1.29	22.2	0.1362	1.85	22.3	0.2565	1.85	22.8	0.4597	1.87
T204-3		B-2A Spider Creek	9098	6.6	0.0188	94.5	0.032	1.58	136.8	0.0419	1.68	163.8	0.0641	1.74
T195-1		5 Pinedale	12 159.6	9	0.0111	137.6	0.022	1.93	167	0.0343	2	193.4	0.0543	2.06
E894-2		1 Old Road	11 927.8	11.3	0.00925	70	0.0432	1.89	74	0.0775	1.91	93.7	0.1121	2.02
E712-2		B-55 Big Piney	3480.8	8.9	0.0058	76.8	0.0393	1.75	106.6	0.0538	1.88	119.9	0.0876	1.92
B029-3		A-1 Wasp	11 443.8	6.6	0.00178	298.3	0.0101	2.05	329.2	0.0174	2.08	382.6	0.0274	2.14
SHELL Vible		Vible 1D-11D	12 518.5	5.9	0.00172	215.1	0.014	1.85	270.3	0.0212	1.93	339.8	0.0309	2
E894-1		1 Old Road	11 897.3	5.2	0.000995	152.6	0.0198	1.65	222.1	0.0258	1.78	292.4	0.0359	1.87
B029-1		A-1 Wasp	11 332.9	3.6	0.000832	258.4	0.0117	1.63	327.8	0.0175	1.7	443.5	0.0237	1.79
R780-3		C-47 Tip Top Shallow	2759.9	9.1	0.000708	87.3	0.0346	1.84	101.7	0.0563	1.91	142.1	0.0739	2.04
T195-2		5 Pinedale	12 161.5	6.5	0.000361	249.4	0.0121	1.99	298.2	0.0192	2.06	361.4	0.0291	2.13
B029-4		A-1 Wasp	11 534	1.5	0.000137	195.2	0.0155	1.21	280.6	0.0204	1.29	351	0.0299	1.35
B029-2		A-1 Wasp	11 374.9	0.5	0.000018	301.8	0.01	0.99	490.1	0.0117	1.08	753.8	0.0139	1.15

Table 3.

Experimental data of rock resistivity in the Piceance Basin

				Brine salinity (Kppm):		20			40			80
				C_w (mhos):		3.02			5.73			10.5
				R_w (⁠ $Ω$ ·m):		0.331			0.17452			0.09524
USGS Library No.	Basin	Well name	Depth (ft)	Routine porosity (%)	In situ Klinkenberg gas permeability (mD)	R₀/R_w	1/R₀	m, a = 1	R₀/R_w	1/R₀	m, a = 1	R₀/R_w	1/R₀	m, a = 1
E458	Piceance	M-30-2-96W /D-037934	6509.4	10.2	0.0269	87.2	0.0347	1.96	93.4	0.0614	1.99	114	0.0921	2.08
PA424		Williams PA-424-40	4696.5	10.2	0.00307	54.1	0.0559	1.75	72.9	0.0786	1.88	84.7	0.124	1.94
PA424		Williams PA-424-50	6643.5	9	0.00463	64.5	0.0468	1.73	74.5	0.077	1.79	90.4	0.1162	1.87
PA424		Williams PA-424-45	6146.5	8.9	0.0058	64.1	0.0471	1.72	81	0.0707	1.81	106.3	0.0988	1.93
E458		M-30-2-96W /D-037934	6508.3	8.2	0.018	97.5	0.031	1.83	111.4	0.0514	1.88	120.4	0.0872	1.91
PA424		Williams PA-424-42	5142.5	7.6	0.00844	81.2	0.0372	1.71	100.6	0.057	1.79	133.7	0.0785	1.9
PA424		Williams PA-424-44	5192.7	7.1	0.00936	104.9	0.0288	1.76	137.9	0.0416	1.86	156.1	0.0673	1.91
PA424		Williams PA-424-36	4651.6	6.2	0.0134	83.5	0.0362	1.59	124.4	0.0461	1.73	158.4	0.0663	1.82
E436		MV 33-34	6579.8	4.7	0.000247	118.9	0.0254	1.57	157	0.0365	1.66	198.1	0.053	1.73
PA424		Williams PA-424-34	4574.6	4.4	0.00279	154.9	0.0195	1.61	181.4	0.0316	1.66	203.2	0.0517	1.7
E436		MV 33-34	6582	3.6	0.000479	139.6	0.0216	1.48	183.2	0.0313	1.56	254.2	0.0413	1.66
E458		M-30-2-96W /D-037934	6379.5	3.6	0.000303	170.1	0.0178	1.54	214.7	0.0267	1.61	279.6	0.0376	1.69
PA424		Williams PA-424-49	6632.8	3.3	0.00507	130.9	0.0231	1.43	151.1	0.0379	1.47	186.2	0.0564	1.53
E436		MV 33-34	6592.5	2.2	0.000163	128.6	0.0235	1.28	186.4	0.0307	1.38	264.5	0.0397	1.47
PA424		Williams PA-424-35	4635.4	2.1	0.000771	110.9	0.0272	1.22	156.2	0.0367	1.31	228.2	0.046	1.41
B43C		Last Dance 43C-3-792	3577.6	1.7	0.000062	344.6	0.0088	1.43	481.6	0.0119	1.51	90.6	0.0152	1.6
B43C		Last Dance 43C-3-792	3970	0.8	0.000088	449.7	0.0067	1.27	519.7	0.011	1.3	733.3	0.0143	1.38

				Brine salinity (Kppm):		20			40			80
				C_w (mhos):		3.02			5.73			10.5
				R_w (⁠ $Ω$ ·m):		0.331			0.17452			0.09524
USGS Library No.	Basin	Well name	Depth (ft)	Routine porosity (%)	In situ Klinkenberg gas permeability (mD)	R₀/R_w	1/R₀	m, a = 1	R₀/R_w	1/R₀	m, a = 1	R₀/R_w	1/R₀	m, a = 1
E458	Piceance	M-30-2-96W /D-037934	6509.4	10.2	0.0269	87.2	0.0347	1.96	93.4	0.0614	1.99	114	0.0921	2.08
PA424		Williams PA-424-40	4696.5	10.2	0.00307	54.1	0.0559	1.75	72.9	0.0786	1.88	84.7	0.124	1.94
PA424		Williams PA-424-50	6643.5	9	0.00463	64.5	0.0468	1.73	74.5	0.077	1.79	90.4	0.1162	1.87
PA424		Williams PA-424-45	6146.5	8.9	0.0058	64.1	0.0471	1.72	81	0.0707	1.81	106.3	0.0988	1.93
E458		M-30-2-96W /D-037934	6508.3	8.2	0.018	97.5	0.031	1.83	111.4	0.0514	1.88	120.4	0.0872	1.91
PA424		Williams PA-424-42	5142.5	7.6	0.00844	81.2	0.0372	1.71	100.6	0.057	1.79	133.7	0.0785	1.9
PA424		Williams PA-424-44	5192.7	7.1	0.00936	104.9	0.0288	1.76	137.9	0.0416	1.86	156.1	0.0673	1.91
PA424		Williams PA-424-36	4651.6	6.2	0.0134	83.5	0.0362	1.59	124.4	0.0461	1.73	158.4	0.0663	1.82
E436		MV 33-34	6579.8	4.7	0.000247	118.9	0.0254	1.57	157	0.0365	1.66	198.1	0.053	1.73
PA424		Williams PA-424-34	4574.6	4.4	0.00279	154.9	0.0195	1.61	181.4	0.0316	1.66	203.2	0.0517	1.7
E436		MV 33-34	6582	3.6	0.000479	139.6	0.0216	1.48	183.2	0.0313	1.56	254.2	0.0413	1.66
E458		M-30-2-96W /D-037934	6379.5	3.6	0.000303	170.1	0.0178	1.54	214.7	0.0267	1.61	279.6	0.0376	1.69
PA424		Williams PA-424-49	6632.8	3.3	0.00507	130.9	0.0231	1.43	151.1	0.0379	1.47	186.2	0.0564	1.53
E436		MV 33-34	6592.5	2.2	0.000163	128.6	0.0235	1.28	186.4	0.0307	1.38	264.5	0.0397	1.47
PA424		Williams PA-424-35	4635.4	2.1	0.000771	110.9	0.0272	1.22	156.2	0.0367	1.31	228.2	0.046	1.41
B43C		Last Dance 43C-3-792	3577.6	1.7	0.000062	344.6	0.0088	1.43	481.6	0.0119	1.51	90.6	0.0152	1.6
B43C		Last Dance 43C-3-792	3970	0.8	0.000088	449.7	0.0067	1.27	519.7	0.011	1.3	733.3	0.0143	1.38

Table 4.

Rock resistivity (R₀) analysis of saturated formation water in the Green River Basin

No.	Well name	Depth (ft)	Routine porosity (%)	In situ Klinkenberg gas permeability (mD)	Experimental R₀	Archie porosity exponent, m in situ	Experimental R_w	Calculated R_x	Calculated F_d	Calculated R₀
1	Vible 1D-11D	12 529	1.4	0.000025	96.1538	1.4200	0.1745	60.1019	184.3492	92.2745
2	A-1 Wasp	10 537.2	3.5	0.00019	44.0529	1.6200	0.1745	26.3267	138.7802	50.5467
3	A-1 Wasp	11 332.9	3.5	0.000728	50.0000	1.6500	0.1745	15.2392	114.9885	35.3070
4	A-1 Wasp	11 592.7	4.7	0.000227	26.5957	1.6200	0.1745	24.4877	135.3659	48.1117
5	A-1 Wasp	11 459.1	5.4	0.000827	45.0450	1.8300	0.1745	14.4686	112.9541	34.1813
6	A-1 Wasp	11 548	5.8	0.001	29.4985	1.7700	0.1745	13.3921	109.9899	32.5876
7	1 Old Road	11 915.2	8.4	0.00372	21.3675	1.9000	0.1745	7.8458	91.5118	23.8165
8	K-2 Mason	9397.3	8.4	0.000358	29.5858	2.0600	0.1745	20.3432	127.0016	42.5075
9	K-2 Mason	7703.8	12.1	0.2386	11.1483	1.9000	0.1745	1.4426	51.1068	10.3617
10	B-54 Big Piney	3511.8	13.1	0.7063	11.0497	1.9600	0.1745	0.9275	43.9030	8.5895
11	B-54 Big Piney	3403.9	16.7	1.2	6.4267	1.9200	0.1745	0.7475	40.7631	7.8615
12	B-54 Big Piney	3433.9	17.2	24.5	5.1733	1.8100	0.1745	0.2190	26.7220	4.8825
13	C-47 Tip Top Shallow	2783.4	22.3	21.5	3.6982	1.9400	0.1745	0.2310	27.2152	4.9806
14	C-47 Tip Top Shallow	2831.8	23.6	2.73	3.9793	2.0800	0.1745	0.5350	36.3321	6.8756

No.	Well name	Depth (ft)	Routine porosity (%)	In situ Klinkenberg gas permeability (mD)	Experimental R₀	Archie porosity exponent, m in situ	Experimental R_w	Calculated R_x	Calculated F_d	Calculated R₀
1	Vible 1D-11D	12 529	1.4	0.000025	96.1538	1.4200	0.1745	60.1019	184.3492	92.2745
2	A-1 Wasp	10 537.2	3.5	0.00019	44.0529	1.6200	0.1745	26.3267	138.7802	50.5467
3	A-1 Wasp	11 332.9	3.5	0.000728	50.0000	1.6500	0.1745	15.2392	114.9885	35.3070
4	A-1 Wasp	11 592.7	4.7	0.000227	26.5957	1.6200	0.1745	24.4877	135.3659	48.1117
5	A-1 Wasp	11 459.1	5.4	0.000827	45.0450	1.8300	0.1745	14.4686	112.9541	34.1813
6	A-1 Wasp	11 548	5.8	0.001	29.4985	1.7700	0.1745	13.3921	109.9899	32.5876
7	1 Old Road	11 915.2	8.4	0.00372	21.3675	1.9000	0.1745	7.8458	91.5118	23.8165
8	K-2 Mason	9397.3	8.4	0.000358	29.5858	2.0600	0.1745	20.3432	127.0016	42.5075
9	K-2 Mason	7703.8	12.1	0.2386	11.1483	1.9000	0.1745	1.4426	51.1068	10.3617
10	B-54 Big Piney	3511.8	13.1	0.7063	11.0497	1.9600	0.1745	0.9275	43.9030	8.5895
11	B-54 Big Piney	3403.9	16.7	1.2	6.4267	1.9200	0.1745	0.7475	40.7631	7.8615
12	B-54 Big Piney	3433.9	17.2	24.5	5.1733	1.8100	0.1745	0.2190	26.7220	4.8825
13	C-47 Tip Top Shallow	2783.4	22.3	21.5	3.6982	1.9400	0.1745	0.2310	27.2152	4.9806
14	C-47 Tip Top Shallow	2831.8	23.6	2.73	3.9793	2.0800	0.1745	0.5350	36.3321	6.8756

Table 5.

Rock resistivity (R₀) analysis of saturated formation water in the Piceance Basin

No.	Well name	Depth (ft)	Routine porosity (%)	In situ Klinkenberg gas permeability (mD)	Experimental R₀	Archie porosity exponent, m in situ	Experimental R_w	Calculated R_x	Calculated F_d	Calculated R₀
1	MV 33-34	6591.9	2.4	0.000711	29.2398	1.3600	0.1745	13.1374	100.1200	30.6103
2	T63X-2G	10 636.3	2.5	0.000038	70.4225	1.6100	0.1745	35.8785	162.8123	64.2925
3	MWX-2	7141.9	3.9	0.000065	24.3902	1.5100	0.1745	29.8449	148.9318	55.8365
4	MV 33-34	6579.5	5.8	0.000363	27.0270	1.7300	0.1745	16.5444	111.9404	36.0803
5	MV 33-34	6579.5	7.5	0.000441	19.6464	1.8000	0.1745	15.4759	108.3813	34.3907
6	MWX-2	4939.8	9	0.00426	15.9236	1.8400	0.1745	7.1091	74.3788	20.0897
7	Williams PA-424-46	6148.6	9.9	0.00761	16.9205	1.9300	0.1745	5.8262	67.5494	17.6149
8	MWX-2	7133.5	10.2	0.0067	15.2439	1.9200	0.1745	6.0863	68.9926	18.1269
9	21011-5 Moon Lake	816.5	10.6	0.0205	8.7566	1.6400	0.1745	4.1473	57.3034	14.1479
10	21011-5 Moon Lake	812.6	18.4	17.2	3.8023	1.7700	0.1745	0.4120	18.7428	3.6830

No.	Well name	Depth (ft)	Routine porosity (%)	In situ Klinkenberg gas permeability (mD)	Experimental R₀	Archie porosity exponent, m in situ	Experimental R_w	Calculated R_x	Calculated F_d	Calculated R₀
1	MV 33-34	6591.9	2.4	0.000711	29.2398	1.3600	0.1745	13.1374	100.1200	30.6103
2	T63X-2G	10 636.3	2.5	0.000038	70.4225	1.6100	0.1745	35.8785	162.8123	64.2925
3	MWX-2	7141.9	3.9	0.000065	24.3902	1.5100	0.1745	29.8449	148.9318	55.8365
4	MV 33-34	6579.5	5.8	0.000363	27.0270	1.7300	0.1745	16.5444	111.9404	36.0803
5	MV 33-34	6579.5	7.5	0.000441	19.6464	1.8000	0.1745	15.4759	108.3813	34.3907
6	MWX-2	4939.8	9	0.00426	15.9236	1.8400	0.1745	7.1091	74.3788	20.0897
7	Williams PA-424-46	6148.6	9.9	0.00761	16.9205	1.9300	0.1745	5.8262	67.5494	17.6149
8	MWX-2	7133.5	10.2	0.0067	15.2439	1.9200	0.1745	6.0863	68.9926	18.1269
9	21011-5 Moon Lake	816.5	10.6	0.0205	8.7566	1.6400	0.1745	4.1473	57.3034	14.1479
10	21011-5 Moon Lake	812.6	18.4	17.2	3.8023	1.7700	0.1745	0.4120	18.7428	3.6830

The core samples chosen for this study have undergone electrical resistivity measurement experiments and are representative regarding geographical location, lithology, porosity and permeability. To evaluate the possible W–S cation-exchange effects, data analyses were conducted at different NaCl concentrations of 20 (n = 138), 40 (n = 310) and 80 Kppm (n = 198). Obtaining a known salinity in low-permeability rocks can be challenging due to their low brine permeability, which makes it difficult to achieve flow-through displacement. To address this challenge, the samples were dried at 70°C for several hours and then immersed in a methyl alcohol bath for 24–48 h. The samples were then extracted with methyl alcohol for 3 days, dried and vacuum/pressure saturated with methyl alcohol before being immersed in the methyl alcohol bath for a minimum of 3 days. The samples were then dried at 70°C in a convection oven for not less than 24 h and weighed to confirm that the sample weights had returned to their original pre-saturation weights for clean, dry samples. The samples were then vacuum/pressure saturated with brine of different salinity level and left immersed for 2–8 weeks. This process was repeated for each change in salinity (Byrne et al. 2008).

Through our analysis, we have verified the existence of additional resistivity (R_x) by fitting the relationship between the rock resistivity (⁠ $Ω$ ·m) of fully saturated brine (R₀) and formation water resistivity (R_w) (Figs 8 and 9).

Fig. 8.

(a)–(c) Fitting diagrams for fully saturated rock resistivity (R0) and formation water resistivity (Rw) core with different permeabilities (mD) (Green River Basin).

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(a)–(c) Fitting diagrams for fully saturated rock resistivity (R₀) and formation water resistivity (R_w) core with different permeabilities (mD) (Green River Basin).

Fig. 9.

(a)–(c) Fitting diagrams for fully saturated rock resistivity (R0) and formation water resistivity (Rw) core with different permeabilities (mD) (Piceance Basin).

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(a)–(c) Fitting diagrams for fully saturated rock resistivity (R₀) and formation water resistivity (R_w) core with different permeabilities (mD) (Piceance Basin).

Figures 8 and 9 provide clear evidence of the variation in formation water resistivity (R_w) and saturation across different permeability cores in the Green River Basin and the Piceance Basin, spanning several orders of magnitude. The strong linear relationship observed in the data for formation water resistivity (R₀) confirms the accuracy of equation (3) and the existence of additional resistivity (R_x). Analysis of the core images suggests that the dominant factors influencing additional resistivity (R_x) are pore structure – that is, porosity and pore connectivity. It is important to note that additional resistivity (R_x) is always present and does not reach 0.

Discussion

To better understand the relationship between the resistivity change factor (F_d) and the additional resistivity (R_x) with rock pore structure, we conducted a fitting analysis in this study. Specifically, we analysed the relationship between the formation resistivity change factor (F_d) with permeability (K), as well as the relationship between resistivity change factor (F_d) and porosity (⁠ $Φ$ ⁠), and analysed the relationship between R_x and K and the relationship between R_x and porosity (⁠ $Φ$ ⁠) (Figs 8–11).

Fig. 10.

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Fitting relationship between: (a) the formation resistivity change factor (F_d) and permeability (K); and (b) additional resistivity (R_x) and permeability (K) (Green River Basin).

Fig. 11.

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Fitting relationship between: (a) the formation resistivity change factor (F_d) and porosity (⁠ $Φ$ ⁠); and (b) additional resistivity (R_x) and porosity (⁠ $Φ$ ⁠) (Green River Basin).

The results of the fitting equations demonstrated a strong linear relationship between R_x and K when plotted on a log–log coordinate for the core data in both basins. This suggests that the fluid-flow channels within the rock pores play a significantly role in the additional resistivity (R_x). Furthermore, the relationship F_d and K is stronger than that between F_d and $Φ$ in the Green River Basin, while in the Piceance Basin the relationship F_d and $Φ$ appears to be stronger than the one between F_d and K. These findings suggest that Piceance Basin cores have a lower porosity and permeability compared to those of the Green River Basin. It can be inferred that when K is below 10⁻³ mD, the resistivity change factor (F_d) is primarily influenced by porosity (⁠ $Φ$ ⁠), based on the core data and images from both basins.

Figures 10–13 demonstrate a negative correlation between the values of additional resistivity (R_x) and the formation resistivity change factor (F_d) with permeability (K). R_x represents the equivalent resistivity added by pore networks in series with pore expansion, and when pore connectivity is optimal the corresponding additional resistivity is low. As such, the underlying physical meaning is aligned with the observed trend. As F_d and R_x exhibit a strong correlation with K in the two basins, this relationship can be utilized within the fitting formula to calculate the F_d and R_x values of a given core permeability. This information can then be used in equation (3) to estimate the rock resistivity (R₀) fully saturated by brine.

Fig. 12.

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Fitting relationship between: (a) the formation resistivity change factor (F_d) and permeability (K); and (b) additional resistivity (R_x) and permeability (K) (Piceance Basin).

Fig. 13.

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Fitting relationship between: (a) the formation resistivity change factor (F_d) and porosity (⁠ $Φ$ ⁠); and (b) additional resistivity (R_x) and porosity (⁠ $Φ$ ⁠) (Piceance Basin).

The fitting formula is as follows:

in the Green River Basin:

R_{x} = 0.8051 K^{- 0.407}

(26)

F_{d} = 41.817 K^{- 0.14}

(27)

in the Piceance Basin:

R_{x} = 1.0932 K^{- 0.343}

(28)

F_{d} = 30.056 K^{- 0.166}

(29)

where R_x is additional resistivity (⁠

Ω

·m), F_d is the formation resistivity change factor and K is permeability (mD).

We conducted a new analysis using experimental data from sandstones with varying permeability in both the Green River and Piceance basins to validate the general applicability of our fitting formula (Tables 4 and 5). We used the fitting formulae (equations 26–29) and equation (3) to calculate the rock resistivity (⁠ $Ω$ ·m) of fully saturated brine (R₀). We then compared the estimated values for saturated formation water (R₀) with the laboratory measurements and found them to be very similar (Figs 14 and 15).

Fig. 14.

Correlation analysis between the calculated values of rock resistivity of saturated formation water (R0) and the measured values in the laboratory (Green River Basin).

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Correlation analysis between the calculated values of rock resistivity of saturated formation water (R₀) and the measured values in the laboratory (Green River Basin).

Fig. 15.

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Correlation analysis between the calculated values of rock resistivity of saturated formation water (R₀) and the measured values in the laboratory (Piceance Basin).

The proposed method of calculation has provided a new approach for estimating the rock resistivity of saturated water (R₀) using permeability. In practical scenarios, the rock resistivity (⁠ $Ω$ ·m) of fully saturated brine (R₀) may not be readily available for all core samples. As a result, the fitting formula and equation (3) can be used to obtain an estimation of the rock resistivity (⁠ $Ω$ ·m) of fully saturated brine (R₀) within a given basin.

Conclusions

In this study, we investigated the presence of additional resistivity (R_x) through theoretical and data analysis. Our results indicate that the pore structure of the material, particularly porosity and pore connectivity, is the primarily influence on additional resistivity (R_x). Furthermore, our findings suggest that additional resistivity (R_x) is consistently present and cannot be reduced to 0.

In addition, our study demonstrated the relationship between additional resistivity (R_x), the formation resistivity change factor (F_d) and permeability (K) through an analysis of experimental data. When plotted in logarithmic coordinates, our results revealed a strong linear correlation between the formation resistivity change factor (F_d) and additional resistivity (R_x) with respect to permeability (K). Specifically, we observed a decline in both the formation resistivity change factor (F_d) and additional resistivity (R_x) as permeability (K) increased, which is consistent with our theoretical expectations.

To demonstrate the broad applicability of our fitting formula, we utilized experimental data from a sandstone reservoir in a basin in the western USA (Green River) with varying permeability levels. We applied our fitting formula to calculate the rock resistivity of saturated formation water (R₀). Our results showed that the estimated values for the rock resistivity (⁠ $Ω$ ·m) of fully saturated brine (R_o) were remarkably similar to the laboratory measurements. Therefore, our study has provided a new approach for calculating resistivity (R₀) of rock in saturated formation water.

Author contributions

WZ: writing – original draft (lead); TL: writing – review & editing (supporting); JT: writing – review & editing (equal); JZ: writing – review & editing (supporting)

Funding

This work was funded by the National 13th 5-Year Plan of Oil and Gas Program of China (2017ZX05019-001), National Natural Science Foundation of China (41476027), PetroChina Key Technological Program (2016E-0503) and CNPC Research Projec (2017 F-16).

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

All data generated or analysed during this study are included in this published article (and, if present, its supplementary information files).

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This is an Open Access article distributed under the terms of the Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0/)

Estimating additional resistivity by permeability in brine-saturated sandstones

Abstract

Geological setting

Methodology

Relationship between the permeability of the curved capillary model and the formation factor

Relationship between permeability and the formation factor in a complex curved capillary model

Results

Discussion

Conclusions

Author contributions

Funding

Competing interests

Data availability

References

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Estimating additional resistivity by permeability in brine-saturated sandstones

Abstract

Geological setting

Methodology

Relationship between the permeability of the curved capillary model and the formation factor

Relationship between permeability and the formation factor in a complex curved capillary model

Results

Discussion

Conclusions

Author contributions

Funding

Competing interests

Data availability

References

Email alerts

Citing articles via

About GSW

Resources

Explore

Connect

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